Overview
This lecture covers derived Boolean operators (NAND, NOR, XOR, XNOR), their representation in Java and databases, and how logic differs in three-valued (ternary) systems.
Derived Boolean Operators
- NAND is NOT-AND; true except when both inputs are true.
- NOR is NOT-OR; true only when both inputs are false.
- XOR (exclusive OR) is true when inputs differ; false when both are the same.
- XNOR (equivalence) is true when inputs are the same; false otherwise.
- Derived operators have special symbols and gates (e.g., NAND: AND gate with circle, XOR: OR gate with extra line).
- All derived operators can be written using AND, OR, and NOT.
Boolean Operators in Java
- Java uses
| (bitwise OR), & (bitwise AND), ^ (XOR), ! (NOT), || (Boolean OR), and && (Boolean AND).
- Single
| and & evaluate both sides always; double || and && short-circuit evaluation.
- Short-circuiting avoids unnecessary operations and errors (e.g., division by zero).
- Bitwise operators compare each corresponding bit of two integers.
Logic Simplification & Coding
- Simplifying Boolean expressions improves code readability and efficiency.
- Use Boolean algebra theorems (e.g., De Morgan's, absorption) for simplification.
- Always list theorems used for each simplification step.
Boolean Logic in Databases (Ternary Logic)
- Databases may use three logical values: true, false, and unknown (null).
- NOT unknown remains unknown; AND/OR with unknown follows special truth tables.
- Boolean logic must address cases where data is missing (null).
Key Terms & Definitions
- NAND — NOT of AND; outputs false only if all inputs are true.
- NOR — NOT of OR; outputs true only if all inputs are false.
- XOR — Exclusive OR; outputs true if inputs differ.
- XNOR — Exclusive NOR (equivalence); outputs true if inputs are the same.
- Bitwise Operator — Evaluates logic on each bit of binary values.
- Short-circuit Evaluation — Stops evaluation as soon as result is certain.
- Ternary Logic — Logic system with three values: true, false, unknown (null).
Action Items / Next Steps
- Practice writing and simplifying Boolean expressions.
- Review Java Boolean and bitwise operator syntax.
- Experiment with logic operators in Java or another language.
- Be aware of ternary logic cases when working with databases.